On the propagation of maximally dissipative phase boundaries in solids
Authors:
Rohan Abeyaratne and James K. Knowles
Journal:
Quart. Appl. Math. 50 (1992), 149-172
MSC:
Primary 73B30
DOI:
https://doi.org/10.1090/qam/1146630
MathSciNet review:
MR1146630
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Abstract: This paper is concerned with the kinetics of propagating phase boundaries in a bar made of a special nonlinearly elastic material. First, it is shown that there is a kinetic law of the form $f = \varphi \left ( {\dot s} \right )$ relating the driving traction $f$ at a phase boundary to the phase boundary velocity $\dot s$ that corresponds to a notion of maximum dissipation analogous to the concept of maximum plastic work. Second, it is shown that a modified version of the entropy rate admissibility criterion can be described by a kinetic relation of the above form, but with a different $\varphi$ . Both kinetic relations are applied to the Riemann problem for longitudinal waves in the bar.
- Rohan Abeyaratne and James K. Knowles, Kinetic relations and the propagation of phase boundaries in solids, Arch. Rational Mech. Anal. 114 (1991), no. 2, 119–154. MR 1094433, DOI https://doi.org/10.1007/BF00375400
- O. A. Oleĭnik, On the uniqueness of the generalized solution of the Cauchy problem for a non-linear system of equations occurring in mechanics, Uspehi Mat. Nauk (N.S.) 12 (1957), no. 6(78), 169–176 (Russian). MR 0094543
- Tai Ping Liu, Uniqueness of weak solutions of the Cauchy problem for general $2\times 2$ conservation laws, J. Differential Equations 20 (1976), no. 2, 369–388. MR 393871, DOI https://doi.org/10.1016/0022-0396%2876%2990114-5
C. M. Dafermos, Discontinuous thermokinetic processes, Appendix 4B of Rational Thermodynamics (C. Truesdell, ed.), Springer-Verlag, New York, 1984
J. W. Christian, The Theory of Transformations in Metals and Alloys, Part I, Pergamon Press, Oxford, 1975
- Rohan Abeyaratne and James K. Knowles, On the dissipative response due to discontinuous strains in bars of unstable elastic material, Internat. J. Solids Structures 24 (1988), no. 10, 1021–1044. MR 974437, DOI https://doi.org/10.1016/0020-7683%2888%2990105-9
- Rohan Abeyaratne and James K. Knowles, On the driving traction acting on a surface of strain discontinuity in a continuum, J. Mech. Phys. Solids 38 (1990), no. 3, 345–360. MR 1051343, DOI https://doi.org/10.1016/0022-5096%2890%2990003-M
J. R. Rice, On the structure of stress-strain relations for time-dependent plastic deformation in metals, J. Appl. Mech. 37, 728–737 (1970)
- J. Lubliner, A maximum-dissipation principle in generalized plasticity, Acta Mech. 52 (1984), no. 3-4, 225–237. MR 765250, DOI https://doi.org/10.1007/BF01179618
- Michael Shearer, Nonuniqueness of admissible solutions of Riemann initial value problems for a system of conservation laws of mixed type, Arch. Rational Mech. Anal. 93 (1986), no. 1, 45–59. MR 822335, DOI https://doi.org/10.1007/BF00250844
- Michael Shearer, Dynamic phase transitions in a van der Waals gas, Quart. Appl. Math. 46 (1988), no. 4, 631–636. MR 973380, DOI https://doi.org/10.1090/S0033-569X-1988-0973380-8
- M. Slemrod, Admissibility criteria for propagating phase boundaries in a van der Waals fluid, Arch. Rational Mech. Anal. 81 (1983), no. 4, 301–315. MR 683192, DOI https://doi.org/10.1007/BF00250857
- M. Slemrod, Dynamics of first order phase transitions, Phase transformations and material instabilities in solids (Madison, Wis., 1983) Publ. Math. Res. Center Univ. Wisconsin, vol. 52, Academic Press, Orlando, FL, 1984, pp. 163–203. MR 802225
L. Truskinovsky, Equilibrium phase interfaces, Soviet Phys. Dokl. 27, 551–553 (1982)
L. Truskinovsky, Structure of an isothermal phase discontinuity, Soviet Phys. Dokl. 30, 945–948 (1985)
- Constantine M. Dafermos, The entropy rate admissibility criterion for solutions of hyperbolic conservation laws, J. Differential Equations 14 (1973), 202–212. MR 328368, DOI https://doi.org/10.1016/0022-0396%2873%2990043-0
- C. M. Dafermos, Hyperbolic systems of conservation laws, Systems of nonlinear partial differential equations (Oxford, 1982) NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., vol. 111, Reidel, Dordrecht, 1983, pp. 25–70. MR 725517
- Harumi Hattori, The Riemann problem for a van der Waals fluid with entropy rate admissibility criterion—isothermal case, Arch. Rational Mech. Anal. 92 (1986), no. 3, 247–263. MR 816624, DOI https://doi.org/10.1007/BF00254828
- Harumi Hattori, The Riemann problem for a van der Waals fluid with entropy rate admissibility criterion. Nonisothermal case, J. Differential Equations 65 (1986), no. 2, 158–174. MR 861514, DOI https://doi.org/10.1016/0022-0396%2886%2990031-8
- Richard D. James, The propagation of phase boundaries in elastic bars, Arch. Rational Mech. Anal. 73 (1980), no. 2, 125–158. MR 556559, DOI https://doi.org/10.1007/BF00258234
- Thomas J. Pence, On the encounter of an acoustic shear pulse with a phase boundary in an elastic material: energy and dissipation, J. Elasticity 26 (1991), no. 2, 95–146. MR 1128426, DOI https://doi.org/10.1007/BF00041218
- Rohan Abeyaratne and James K. Knowles, Implications of viscosity and strain-gradient effects for the kinetics of propagating phase boundaries in solids, SIAM J. Appl. Math. 51 (1991), no. 5, 1205–1221. MR 1127848, DOI https://doi.org/10.1137/0151061
D. A. Porter and K. E. Easterling, Phase Transformations in Metals and Alloys, van Nostrand-Reinhold, New York, 1981
R. Abeyaratne and J. K. Knowles, Kinetic relations and the propagation of phase boundaries in solids, Arch. Rational Mech. Anal. 114, 119–154 (1991)
O. A. Oleinik, On the uniqueness of the generalized solution of the Cauchy problem for a nonlinear system of equations occurring in mechanics, Uspekhi Mat. Nauk (N.S.) 12, no. 6 (78), 169–176 (1957) (Russian)
T. P. Liu, Uniqueness of weak solutions of the Cauchy Problem for general $2 \times 2$ conservation laws, J. Differential Equations 20, 369–388 (1976)
C. M. Dafermos, Discontinuous thermokinetic processes, Appendix 4B of Rational Thermodynamics (C. Truesdell, ed.), Springer-Verlag, New York, 1984
J. W. Christian, The Theory of Transformations in Metals and Alloys, Part I, Pergamon Press, Oxford, 1975
R. Abeyaratne and J. K. Knowles, On the dissipative response due to discontinuous strains in bars of unstable elastic material, Internat. J. Solids and Structures 24, 1021–1044 (1988)
R. Abeyaratne and J. K. Knowles, On the driving traction acting on a surface of strain discontinuity in a continuum, J. Mech. Phys. Solids 38, 345–360 (1990)
J. R. Rice, On the structure of stress-strain relations for time-dependent plastic deformation in metals, J. Appl. Mech. 37, 728–737 (1970)
J. Lubliner, A maximum dissipation principle in generalized plasticity, Acta Mech. 52, 225–237 (1984)
M. Shearer, Nonuniqueness of admissible solutions of Riemann initial value problems for a system of conservation laws of mixed type, Arch. Rational Mech. Anal. 93, 45–69 (1986)
M. Shearer, Dynamic phase transitions in a van der Waals gas, Quart. Appl. Math. 46, 631–636 (1988)
M. Slemrod, Admissibility criteria for propagating phase boundaries in a van der Waals fluid, Arch. Rational Mech. Anal. 81, 301–315 (1983)
M. Slemrod, Dynamics of first order phase transitions, Phase Transformations and Material Instabilities in Solids (M. E. Gurtin, ed.), Academic Press, New York, 1984, pp. 163–203
L. Truskinovsky, Equilibrium phase interfaces, Soviet Phys. Dokl. 27, 551–553 (1982)
L. Truskinovsky, Structure of an isothermal phase discontinuity, Soviet Phys. Dokl. 30, 945–948 (1985)
C. M. Dafermos, The entropy rate admissibility criterion for solutions of hyperbolic conservation laws, J. Differential Equations 14, 202–212 (1973)
C. M. Dafermos, Hyperbolic systems of conservation laws, Systems of Nonlinear Partial Differential Equations (J. M. Ball, ed.), Reidel, Dordrecht, 1983, pp. 25–70
H. Hattori, The Riemann problem for a van der Waals fluid with entropy rate admissibility criterion. Isothermal case, Arch. Rational Mech. Anal. 92, 247–263 (1986)
H. Hattori, The Riemann problem for a van der Waals fluid with entropy rate admissibility criterion. Nonisothermal case, J. Differential Equations 65, 158–174 (1986)
R. D. James, The propagation of phase boundaries in elastic bars, Arch. Rational Mech. Anal. 73, 125–158 (1980)
T. J. Pence, On the encounter of an accoustic shear pulse with a phase boundary in an elastic material: energy and dissipation, J. Elasticity, to appear
R. Abeyaratne and J. K. Knowles, Implications of viscosity and strain gradient effects for the kinetics of propagating phase boundaries in solids, Technical Report No. 11, ONR contract N00014-87-K-0117, April 1990; to appear in SIAM J. Appl. Math.
D. A. Porter and K. E. Easterling, Phase Transformations in Metals and Alloys, van Nostrand-Reinhold, New York, 1981
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Article copyright:
© Copyright 1992
American Mathematical Society